Question 1026695
g'(x) = {{{x^2}}} ==> {{{g(x) = x^3/3 + c}}} for some constant c.

g(4) = -1/4 ==> {{{g(4) = 4^3/3 +c = -1/4}}} ==> c = -259/12.

==> {{{g(x) = x^3/3 - 259/12}}} 

Now f'(x) = -2x ==> {{{f(x) = -x^2 + k}}} for some constant k.

==> {{{f(g(x)) = -(x^3/3 - 259/12)^2 + k}}}

==> {{{df(g(x))/dx = -2x^2(x^3/3 - 259/12)}}}

==> {{{df(g(4))/dx = -2*4^2(4^3/3 - 259/12) = 8}}}